Abstract
Demographic stochasticity, the random fluctuations arising from the intrinsic discreteness of populations and the uncertainty of individual birth and death events, is an essential feature of population dynamics. Nevertheless theoretical investigations often neglect this naturally occurring noise due to the mathematical complexity of stochastic models. This paper reports the results of analytical and computational investigations of models of competitive population dynamics, specifically the competition between species in heterogeneous environments with different phenotypes of dispersal, fully accounting for demographic stochasticity. A novel asymptotic approximation is introduced and applied to derive remarkably simple analytical forms for key statistical quantities describing the populations’ dynamical evolution. These formulas characterize the selection processes that determine which (if either) competitor has an evolutionary advantage. The theory is verified by large-scale numerical simulations. We discover that the fluctuations can (1) break dynamical degeneracies, (2) support polymorphism that does not exist in deterministic models, (3) reverse the direction of the weak selection and cause shifts in selection regimes, and (4) allow for the emergence of evolutionarily stable dispersal rates. Dynamical mechanisms and time scales of the fluctuation-induced phenomena are identified within the theoretical approach.
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Notes
The claim is verified by numerical observations in Fig. 2.
The analytical conclusion is not limited by restriction, however, as verified by numerical simulations in Sect. 4.
The only exceptions being when \(\mu _x\) and \(\mu _Y \ll 1\), as pointed out in previous work (Lin et al. 2013); with such parameters the patches are almost decoupled. The asymptotic analysis breaks down because the separation of time scale no longer holds true.
Details of the derivations are documented in Lin’s dissertation (2013).
Fundamentally there are several distinct model settings between our, Hamilton and May’s (1977), and Comins et al. (1980) models: (1) Hamilton and May’s and Comins’ models have discrete generations; as a consequence, the competition across generations does not exist. On the contrary, the lifespans of individuals overlap in our model and the cross-generation competition is considered. (2) Hamilton and May’s and Comins’ models incorporated dispersal costs, which are absent in our model. (3) There are catastrophic events in Comins’ model, which is again neglected in our study. As a consequence, both Hamilton and May’s and Comins’ models featured non-zero evolutionarily stable dispersal rates (when the population is finite) even in the homogeneous environments—whereas in the first part of the study we showed that there exists no finite evolutionarily stable dispersal rate in in homogeneous environments (Lin et al. 2013).
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Acknowledgments
This work was supported in part by NSF Awards DMS-0927587 and PHY-1205219. One of us (HK) also acknowledges support from the NSF’s Institute for Mathematics and Its Applications.
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Lin, Y.T., Kim, H. & Doering, C.R. Demographic stochasticity and evolution of dispersion II: Spatially inhomogeneous environments. J. Math. Biol. 70, 679–707 (2015). https://doi.org/10.1007/s00285-014-0756-0
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DOI: https://doi.org/10.1007/s00285-014-0756-0
Keywords
- Stochastic dynamics
- Evolutionarily stable dispersal
- Patch dynamics
- Small noise approximation
- Mean-field dynamics